LMFDB - Embedded newform 150.6.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,6,Mod(1,150)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("150.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	150.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$24.0575729719$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	150.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} -233.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} -233.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -498.000 q^{11} +144.000 q^{12} -809.000 q^{13} -932.000 q^{14} +256.000 q^{16} +1002.00 q^{17} +324.000 q^{18} -1705.00 q^{19} -2097.00 q^{21} -1992.00 q^{22} -1554.00 q^{23} +576.000 q^{24} -3236.00 q^{26} +729.000 q^{27} -3728.00 q^{28} +7830.00 q^{29} +977.000 q^{31} +1024.00 q^{32} -4482.00 q^{33} +4008.00 q^{34} +1296.00 q^{36} +4822.00 q^{37} -6820.00 q^{38} -7281.00 q^{39} -8148.00 q^{41} -8388.00 q^{42} -19469.0 q^{43} -7968.00 q^{44} -6216.00 q^{46} -8418.00 q^{47} +2304.00 q^{48} +37482.0 q^{49} +9018.00 q^{51} -12944.0 q^{52} -17664.0 q^{53} +2916.00 q^{54} -14912.0 q^{56} -15345.0 q^{57} +31320.0 q^{58} +35910.0 q^{59} +3527.00 q^{61} +3908.00 q^{62} -18873.0 q^{63} +4096.00 q^{64} -17928.0 q^{66} -57473.0 q^{67} +16032.0 q^{68} -13986.0 q^{69} -7548.00 q^{71} +5184.00 q^{72} +646.000 q^{73} +19288.0 q^{74} -27280.0 q^{76} +116034. q^{77} -29124.0 q^{78} -22720.0 q^{79} +6561.00 q^{81} -32592.0 q^{82} -11574.0 q^{83} -33552.0 q^{84} -77876.0 q^{86} +70470.0 q^{87} -31872.0 q^{88} -78960.0 q^{89} +188497. q^{91} -24864.0 q^{92} +8793.00 q^{93} -33672.0 q^{94} +9216.00 q^{96} -54593.0 q^{97} +149928. q^{98} -40338.0 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	−233.000	−1.79726	−0.898630	−	0.438708i	$-0.855436\pi$
$7$	−233.000	−1.79726	−0.898630	+	0.438708i	$0.855436\pi$
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	0	0
$11$	−498.000	−1.24093	−0.620465	−	0.784234i	$-0.713056\pi$
$11$	−498.000	−1.24093	−0.620465	+	0.784234i	$0.713056\pi$
$12$	144.000	0.288675
$13$	−809.000	−1.32767	−0.663835	−	0.747879i	$-0.731072\pi$
$13$	−809.000	−1.32767	−0.663835	+	0.747879i	$0.731072\pi$
$14$	−932.000	−1.27085
$15$	0	0
$16$	256.000	0.250000
$17$	1002.00	0.840902	0.420451	−	0.907315i	$-0.361872\pi$
$17$	1002.00	0.840902	0.420451	+	0.907315i	$0.361872\pi$
$18$	324.000	0.235702
$19$	−1705.00	−1.08353	−0.541764	−	0.840530i	$-0.682243\pi$
$19$	−1705.00	−1.08353	−0.541764	+	0.840530i	$0.682243\pi$
$20$	0	0
$21$	−2097.00	−1.03765
$22$	−1992.00	−0.877471
$23$	−1554.00	−0.612536	−0.306268	−	0.951945i	$-0.599080\pi$
$23$	−1554.00	−0.612536	−0.306268	+	0.951945i	$0.599080\pi$
$24$	576.000	0.204124
$25$	0	0
$26$	−3236.00	−0.938804
$27$	729.000	0.192450
$28$	−3728.00	−0.898630
$29$	7830.00	1.72889	0.864444	−	0.502729i	$-0.167671\pi$
$29$	7830.00	1.72889	0.864444	+	0.502729i	$0.167671\pi$
$30$	0	0
$31$	977.000	0.182596	0.0912978	−	0.995824i	$-0.470898\pi$
$31$	977.000	0.182596	0.0912978	+	0.995824i	$0.470898\pi$
$32$	1024.00	0.176777
$33$	−4482.00	−0.716452
$34$	4008.00	0.594608
$35$	0	0
$36$	1296.00	0.166667
$37$	4822.00	0.579059	0.289530	−	0.957169i	$-0.406501\pi$
$37$	4822.00	0.579059	0.289530	+	0.957169i	$0.406501\pi$
$38$	−6820.00	−0.766170
$39$	−7281.00	−0.766531
$40$	0	0
$41$	−8148.00	−0.756992	−0.378496	−	0.925603i	$-0.623559\pi$
$41$	−8148.00	−0.756992	−0.378496	+	0.925603i	$0.623559\pi$
$42$	−8388.00	−0.733728
$43$	−19469.0	−1.60573	−0.802865	−	0.596161i	$-0.796692\pi$
$43$	−19469.0	−1.60573	−0.802865	+	0.596161i	$0.796692\pi$
$44$	−7968.00	−0.620465
$45$	0	0
$46$	−6216.00	−0.433128
$47$	−8418.00	−0.555859	−0.277929	−	0.960602i	$-0.589648\pi$
$47$	−8418.00	−0.555859	−0.277929	+	0.960602i	$0.589648\pi$
$48$	2304.00	0.144338
$49$	37482.0	2.23014
$50$	0	0
$51$	9018.00	0.485495
$52$	−12944.0	−0.663835
$53$	−17664.0	−0.863773	−0.431886	−	0.901928i	$-0.642152\pi$
$53$	−17664.0	−0.863773	−0.431886	+	0.901928i	$0.642152\pi$
$54$	2916.00	0.136083
$55$	0	0
$56$	−14912.0	−0.635427
$57$	−15345.0	−0.625576
$58$	31320.0	1.22251
$59$	35910.0	1.34303	0.671514	−	0.740991i	$-0.265644\pi$
$59$	35910.0	1.34303	0.671514	+	0.740991i	$0.265644\pi$
$60$	0	0
$61$	3527.00	0.121361	0.0606807	−	0.998157i	$-0.480673\pi$
$61$	3527.00	0.121361	0.0606807	+	0.998157i	$0.480673\pi$
$62$	3908.00	0.129115
$63$	−18873.0	−0.599087
$64$	4096.00	0.125000
$65$	0	0
$66$	−17928.0	−0.506608
$67$	−57473.0	−1.56414	−0.782072	−	0.623188i	$-0.785837\pi$
$67$	−57473.0	−1.56414	−0.782072	+	0.623188i	$0.785837\pi$
$68$	16032.0	0.420451
$69$	−13986.0	−0.353648
$70$	0	0
$71$	−7548.00	−0.177699	−0.0888497	−	0.996045i	$-0.528319\pi$
$71$	−7548.00	−0.177699	−0.0888497	+	0.996045i	$0.528319\pi$
$72$	5184.00	0.117851
$73$	646.000	0.0141881	0.00709407	−	0.999975i	$-0.497742\pi$
$73$	646.000	0.0141881	0.00709407	+	0.999975i	$0.497742\pi$
$74$	19288.0	0.409457
$75$	0	0
$76$	−27280.0	−0.541764
$77$	116034.	2.23028
$78$	−29124.0	−0.542019
$79$	−22720.0	−0.409582	−0.204791	−	0.978806i	$-0.565651\pi$
$79$	−22720.0	−0.409582	−0.204791	+	0.978806i	$0.565651\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−32592.0	−0.535274
$83$	−11574.0	−0.184412	−0.0922058	−	0.995740i	$-0.529392\pi$
$83$	−11574.0	−0.184412	−0.0922058	+	0.995740i	$0.529392\pi$
$84$	−33552.0	−0.518824
$85$	0	0
$86$	−77876.0	−1.13542
$87$	70470.0	0.998174
$88$	−31872.0	−0.438735
$89$	−78960.0	−1.05665	−0.528326	−	0.849041i	$-0.677180\pi$
$89$	−78960.0	−1.05665	−0.528326	+	0.849041i	$0.677180\pi$
$90$	0	0
$91$	188497.	2.38617
$92$	−24864.0	−0.306268
$93$	8793.00	0.105422
$94$	−33672.0	−0.393051
$95$	0	0
$96$	9216.00	0.102062
$97$	−54593.0	−0.589125	−0.294563	−	0.955632i	$-0.595174\pi$
$97$	−54593.0	−0.589125	−0.294563	+	0.955632i	$0.595174\pi$
$98$	149928.	1.57695
$99$	−40338.0	−0.413644
$100$	0	0
$101$	105552.	1.02959	0.514793	−	0.857314i	$-0.327869\pi$
$101$	105552.	1.02959	0.514793	+	0.857314i	$0.327869\pi$
$102$	36072.0	0.343297
$103$	177436.	1.64797	0.823984	−	0.566613i	$-0.191747\pi$
$103$	177436.	1.64797	0.823984	+	0.566613i	$0.191747\pi$
$104$	−51776.0	−0.469402
$105$	0	0
$106$	−70656.0	−0.610779
$107$	183792.	1.55191	0.775956	−	0.630787i	$-0.217268\pi$
$107$	183792.	1.55191	0.775956	+	0.630787i	$0.217268\pi$
$108$	11664.0	0.0962250
$109$	169685.	1.36797	0.683986	−	0.729495i	$-0.260245\pi$
$109$	169685.	1.36797	0.683986	+	0.729495i	$0.260245\pi$
$110$	0	0
$111$	43398.0	0.334320
$112$	−59648.0	−0.449315
$113$	−263484.	−1.94115	−0.970573	−	0.240808i	$-0.922588\pi$
$113$	−263484.	−1.94115	−0.970573	+	0.240808i	$0.922588\pi$
$114$	−61380.0	−0.442349
$115$	0	0
$116$	125280.	0.864444
$117$	−65529.0	−0.442557
$118$	143640.	0.949665
$119$	−233466.	−1.51132
$120$	0	0
$121$	86953.0	0.539910
$122$	14108.0	0.0858155
$123$	−73332.0	−0.437050
$124$	15632.0	0.0912978
$125$	0	0
$126$	−75492.0	−0.423618
$127$	256912.	1.41343	0.706716	−	0.707497i	$-0.250176\pi$
$127$	256912.	1.41343	0.706716	+	0.707497i	$0.250176\pi$
$128$	16384.0	0.0883883
$129$	−175221.	−0.927069
$130$	0	0
$131$	−219048.	−1.11522	−0.557611	−	0.830103i	$-0.688282\pi$
$131$	−219048.	−1.11522	−0.557611	+	0.830103i	$0.688282\pi$
$132$	−71712.0	−0.358226
$133$	397265.	1.94738
$134$	−229892.	−1.10602
$135$	0	0
$136$	64128.0	0.297304
$137$	−228678.	−1.04093	−0.520467	−	0.853882i	$-0.674242\pi$
$137$	−228678.	−1.04093	−0.520467	+	0.853882i	$0.674242\pi$
$138$	−55944.0	−0.250067
$139$	339740.	1.49145	0.745727	−	0.666252i	$-0.232102\pi$
$139$	339740.	1.49145	0.745727	+	0.666252i	$0.232102\pi$
$140$	0	0
$141$	−75762.0	−0.320925
$142$	−30192.0	−0.125652
$143$	402882.	1.64755
$144$	20736.0	0.0833333
$145$	0	0
$146$	2584.00	0.0100325
$147$	337338.	1.28757
$148$	77152.0	0.289530
$149$	306450.	1.13082	0.565411	−	0.824810i	$-0.308718\pi$
$149$	306450.	1.13082	0.565411	+	0.824810i	$0.308718\pi$
$150$	0	0
$151$	198827.	0.709632	0.354816	−	0.934936i	$-0.384544\pi$
$151$	198827.	0.709632	0.354816	+	0.934936i	$0.384544\pi$
$152$	−109120.	−0.383085
$153$	81162.0	0.280301
$154$	464136.	1.57704
$155$	0	0
$156$	−116496.	−0.383265
$157$	−361283.	−1.16976	−0.584882	−	0.811118i	$-0.698859\pi$
$157$	−361283.	−1.16976	−0.584882	+	0.811118i	$0.698859\pi$
$158$	−90880.0	−0.289618
$159$	−158976.	−0.498699
$160$	0	0
$161$	362082.	1.10089
$162$	26244.0	0.0785674
$163$	−338159.	−0.996901	−0.498450	−	0.866918i	$-0.666097\pi$
$163$	−338159.	−0.996901	−0.498450	+	0.866918i	$0.666097\pi$
$164$	−130368.	−0.378496
$165$	0	0
$166$	−46296.0	−0.130399
$167$	−430248.	−1.19379	−0.596895	−	0.802320i	$-0.703599\pi$
$167$	−430248.	−1.19379	−0.596895	+	0.802320i	$0.703599\pi$
$168$	−134208.	−0.366864
$169$	283188.	0.762708
$170$	0	0
$171$	−138105.	−0.361176
$172$	−311504.	−0.802865
$173$	−603354.	−1.53270	−0.766350	−	0.642424i	$-0.777929\pi$
$173$	−603354.	−1.53270	−0.766350	+	0.642424i	$0.777929\pi$
$174$	281880.	0.705815
$175$	0	0
$176$	−127488.	−0.310233
$177$	323190.	0.775398
$178$	−315840.	−0.747166
$179$	−374370.	−0.873310	−0.436655	−	0.899629i	$-0.643837\pi$
$179$	−374370.	−0.873310	−0.436655	+	0.899629i	$0.643837\pi$
$180$	0	0
$181$	−232423.	−0.527330	−0.263665	−	0.964614i	$-0.584931\pi$
$181$	−232423.	−0.527330	−0.263665	+	0.964614i	$0.584931\pi$
$182$	753988.	1.68728
$183$	31743.0	0.0700681
$184$	−99456.0	−0.216564
$185$	0	0
$186$	35172.0	0.0745443
$187$	−498996.	−1.04350
$188$	−134688.	−0.277929
$189$	−169857.	−0.345883
$190$	0	0
$191$	−846198.	−1.67837	−0.839187	−	0.543843i	$-0.816969\pi$
$191$	−846198.	−1.67837	−0.839187	+	0.543843i	$0.816969\pi$
$192$	36864.0	0.0721688
$193$	155581.	0.300651	0.150326	−	0.988637i	$-0.451968\pi$
$193$	155581.	0.300651	0.150326	+	0.988637i	$0.451968\pi$
$194$	−218372.	−0.416574
$195$	0	0
$196$	599712.	1.11507
$197$	103482.	0.189976	0.0949881	−	0.995478i	$-0.469719\pi$
$197$	103482.	0.189976	0.0949881	+	0.995478i	$0.469719\pi$
$198$	−161352.	−0.292490
$199$	−140425.	−0.251369	−0.125685	−	0.992070i	$-0.540113\pi$
$199$	−140425.	−0.251369	−0.125685	+	0.992070i	$0.540113\pi$
$200$	0	0
$201$	−517257.	−0.903059
$202$	422208.	0.728028
$203$	−1.82439e6	−3.10726
$204$	144288.	0.242748
$205$	0	0
$206$	709744.	1.16529
$207$	−125874.	−0.204179
$208$	−207104.	−0.331917
$209$	849090.	1.34458
$210$	0	0
$211$	−462673.	−0.715431	−0.357716	−	0.933831i	$-0.616444\pi$
$211$	−462673.	−0.715431	−0.357716	+	0.933831i	$0.616444\pi$
$212$	−282624.	−0.431886
$213$	−67932.0	−0.102595
$214$	735168.	1.09737
$215$	0	0
$216$	46656.0	0.0680414
$217$	−227641.	−0.328172
$218$	678740.	0.967302
$219$	5814.00	0.00819152
$220$	0	0
$221$	−810618.	−1.11644
$222$	173592.	0.236400
$223$	735271.	0.990114	0.495057	−	0.868860i	$-0.335147\pi$
$223$	735271.	0.990114	0.495057	+	0.868860i	$0.335147\pi$
$224$	−238592.	−0.317714
$225$	0	0
$226$	−1.05394e6	−1.37260
$227$	−967188.	−1.24579	−0.622897	−	0.782304i	$-0.714044\pi$
$227$	−967188.	−1.24579	−0.622897	+	0.782304i	$0.714044\pi$
$228$	−245520.	−0.312788
$229$	−83695.0	−0.105466	−0.0527328	−	0.998609i	$-0.516793\pi$
$229$	−83695.0	−0.105466	−0.0527328	+	0.998609i	$0.516793\pi$
$230$	0	0
$231$	1.04431e6	1.28765
$232$	501120.	0.611254
$233$	873876.	1.05453	0.527266	−	0.849700i	$-0.323217\pi$
$233$	873876.	1.05453	0.527266	+	0.849700i	$0.323217\pi$
$234$	−262116.	−0.312935
$235$	0	0
$236$	574560.	0.671514
$237$	−204480.	−0.236472
$238$	−933864.	−1.06866
$239$	−1.06056e6	−1.20099	−0.600497	−	0.799627i	$-0.705030\pi$
$239$	−1.06056e6	−1.20099	−0.600497	+	0.799627i	$0.705030\pi$
$240$	0	0
$241$	−756823.	−0.839367	−0.419683	−	0.907671i	$-0.637859\pi$
$241$	−756823.	−0.839367	−0.419683	+	0.907671i	$0.637859\pi$
$242$	347812.	0.381774
$243$	59049.0	0.0641500
$244$	56432.0	0.0606807
$245$	0	0
$246$	−293328.	−0.309041
$247$	1.37934e6	1.43857
$248$	62528.0	0.0645573
$249$	−104166.	−0.106470
$250$	0	0
$251$	−635148.	−0.636342	−0.318171	−	0.948033i	$-0.603069\pi$
$251$	−635148.	−0.636342	−0.318171	+	0.948033i	$0.603069\pi$
$252$	−301968.	−0.299543
$253$	773892.	0.760115
$254$	1.02765e6	0.999448
$255$	0	0
$256$	65536.0	0.0625000
$257$	−30708.0	−0.0290014	−0.0145007	−	0.999895i	$-0.504616\pi$
$257$	−30708.0	−0.0290014	−0.0145007	+	0.999895i	$0.504616\pi$
$258$	−700884.	−0.655537
$259$	−1.12353e6	−1.04072
$260$	0	0
$261$	634230.	0.576296
$262$	−876192.	−0.788581
$263$	189516.	0.168949	0.0844747	−	0.996426i	$-0.473079\pi$
$263$	189516.	0.168949	0.0844747	+	0.996426i	$0.473079\pi$
$264$	−286848.	−0.253304
$265$	0	0
$266$	1.58906e6	1.37701
$267$	−710640.	−0.610059
$268$	−919568.	−0.782072
$269$	1.10997e6	0.935256	0.467628	−	0.883925i	$-0.345109\pi$
$269$	1.10997e6	0.935256	0.467628	+	0.883925i	$0.345109\pi$
$270$	0	0
$271$	211952.	0.175313	0.0876565	−	0.996151i	$-0.472062\pi$
$271$	211952.	0.175313	0.0876565	+	0.996151i	$0.472062\pi$
$272$	256512.	0.210226
$273$	1.69647e6	1.37765
$274$	−914712.	−0.736051
$275$	0	0
$276$	−223776.	−0.176824
$277$	−1.04741e6	−0.820198	−0.410099	−	0.912041i	$-0.634506\pi$
$277$	−1.04741e6	−0.820198	−0.410099	+	0.912041i	$0.634506\pi$
$278$	1.35896e6	1.05462
$279$	79137.0	0.0608652
$280$	0	0
$281$	34002.0	0.0256885	0.0128442	−	0.999918i	$-0.495911\pi$
$281$	34002.0	0.0256885	0.0128442	+	0.999918i	$0.495911\pi$
$282$	−303048.	−0.226928
$283$	281101.	0.208639	0.104320	−	0.994544i	$-0.466733\pi$
$283$	281101.	0.208639	0.104320	+	0.994544i	$0.466733\pi$
$284$	−120768.	−0.0888497
$285$	0	0
$286$	1.61153e6	1.16499
$287$	1.89848e6	1.36051
$288$	82944.0	0.0589256
$289$	−415853.	−0.292884
$290$	0	0
$291$	−491337.	−0.340132
$292$	10336.0	0.00709407
$293$	1.55851e6	1.06057	0.530285	−	0.847819i	$-0.322085\pi$
$293$	1.55851e6	1.06057	0.530285	+	0.847819i	$0.322085\pi$
$294$	1.34935e6	0.910452
$295$	0	0
$296$	308608.	0.204728
$297$	−363042.	−0.238817
$298$	1.22580e6	0.799611
$299$	1.25719e6	0.813245
$300$	0	0
$301$	4.53628e6	2.88591
$302$	795308.	0.501785
$303$	949968.	0.594432
$304$	−436480.	−0.270882
$305$	0	0
$306$	324648.	0.198203
$307$	839917.	0.508616	0.254308	−	0.967123i	$-0.418152\pi$
$307$	839917.	0.508616	0.254308	+	0.967123i	$0.418152\pi$
$308$	1.85654e6	1.11514
$309$	1.59692e6	0.951455
$310$	0	0
$311$	−292698.	−0.171601	−0.0858003	−	0.996312i	$-0.527345\pi$
$311$	−292698.	−0.171601	−0.0858003	+	0.996312i	$0.527345\pi$
$312$	−465984.	−0.271010
$313$	−127859.	−0.0737684	−0.0368842	−	0.999320i	$-0.511743\pi$
$313$	−127859.	−0.0737684	−0.0368842	+	0.999320i	$0.511743\pi$
$314$	−1.44513e6	−0.827148
$315$	0	0
$316$	−363520.	−0.204791
$317$	−648048.	−0.362209	−0.181104	−	0.983464i	$-0.557967\pi$
$317$	−648048.	−0.362209	−0.181104	+	0.983464i	$0.557967\pi$
$318$	−635904.	−0.352634
$319$	−3.89934e6	−2.14543
$320$	0	0
$321$	1.65413e6	0.895997
$322$	1.44833e6	0.778444
$323$	−1.70841e6	−0.911141
$324$	104976.	0.0555556
$325$	0	0
$326$	−1.35264e6	−0.704915
$327$	1.52716e6	0.789799
$328$	−521472.	−0.267637
$329$	1.96139e6	0.999022
$330$	0	0
$331$	−3.39315e6	−1.70229	−0.851144	−	0.524933i	$-0.824090\pi$
$331$	−3.39315e6	−1.70229	−0.851144	+	0.524933i	$0.824090\pi$
$332$	−185184.	−0.0922058
$333$	390582.	0.193020
$334$	−1.72099e6	−0.844137
$335$	0	0
$336$	−536832.	−0.259412
$337$	1.62085e6	0.777441	0.388720	−	0.921356i	$-0.372917\pi$
$337$	1.62085e6	0.777441	0.388720	+	0.921356i	$0.372917\pi$
$338$	1.13275e6	0.539316
$339$	−2.37136e6	−1.12072
$340$	0	0
$341$	−486546.	−0.226589
$342$	−552420.	−0.255390
$343$	−4.81728e6	−2.21088
$344$	−1.24602e6	−0.567711
$345$	0	0
$346$	−2.41342e6	−1.08378
$347$	1.28638e6	0.573517	0.286758	−	0.958003i	$-0.407422\pi$
$347$	1.28638e6	0.573517	0.286758	+	0.958003i	$0.407422\pi$
$348$	1.12752e6	0.499087
$349$	−2.73055e6	−1.20001	−0.600007	−	0.799994i	$-0.704836\pi$
$349$	−2.73055e6	−1.20001	−0.600007	+	0.799994i	$0.704836\pi$
$350$	0	0
$351$	−589761.	−0.255510
$352$	−509952.	−0.219368
$353$	2.13649e6	0.912564	0.456282	−	0.889835i	$-0.349181\pi$
$353$	2.13649e6	0.912564	0.456282	+	0.889835i	$0.349181\pi$
$354$	1.29276e6	0.548289
$355$	0	0
$356$	−1.26336e6	−0.528326
$357$	−2.10119e6	−0.872561
$358$	−1.49748e6	−0.617523
$359$	3.26406e6	1.33666	0.668332	−	0.743863i	$-0.267009\pi$
$359$	3.26406e6	1.33666	0.668332	+	0.743863i	$0.267009\pi$
$360$	0	0
$361$	430926.	0.174034
$362$	−929692.	−0.372879
$363$	782577.	0.311717
$364$	3.01595e6	1.19308
$365$	0	0
$366$	126972.	0.0495456
$367$	4.15078e6	1.60866	0.804330	−	0.594183i	$-0.202524\pi$
$367$	4.15078e6	1.60866	0.804330	+	0.594183i	$0.202524\pi$
$368$	−397824.	−0.153134
$369$	−659988.	−0.252331
$370$	0	0
$371$	4.11571e6	1.55242
$372$	140688.	0.0527108
$373$	2.14242e6	0.797320	0.398660	−	0.917099i	$-0.369475\pi$
$373$	2.14242e6	0.797320	0.398660	+	0.917099i	$0.369475\pi$
$374$	−1.99598e6	−0.737867
$375$	0	0
$376$	−538752.	−0.196526
$377$	−6.33447e6	−2.29539
$378$	−679428.	−0.244576
$379$	−1.94699e6	−0.696253	−0.348126	−	0.937448i	$-0.613182\pi$
$379$	−1.94699e6	−0.696253	−0.348126	+	0.937448i	$0.613182\pi$
$380$	0	0
$381$	2.31221e6	0.816046
$382$	−3.38479e6	−1.18679
$383$	−2.65052e6	−0.923283	−0.461641	−	0.887067i	$-0.652739\pi$
$383$	−2.65052e6	−0.923283	−0.461641	+	0.887067i	$0.652739\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	622324.	0.212593
$387$	−1.57699e6	−0.535243
$388$	−873488.	−0.294563
$389$	282540.	0.0946686	0.0473343	−	0.998879i	$-0.484927\pi$
$389$	282540.	0.0946686	0.0473343	+	0.998879i	$0.484927\pi$
$390$	0	0
$391$	−1.55711e6	−0.515083
$392$	2.39885e6	0.788474
$393$	−1.97143e6	−0.643873
$394$	413928.	0.134333
$395$	0	0
$396$	−645408.	−0.206822
$397$	2.62066e6	0.834515	0.417257	−	0.908788i	$-0.362991\pi$
$397$	2.62066e6	0.834515	0.417257	+	0.908788i	$0.362991\pi$
$398$	−561700.	−0.177745
$399$	3.57538e6	1.12432
$400$	0	0
$401$	−286248.	−0.0888959	−0.0444479	−	0.999012i	$-0.514153\pi$
$401$	−286248.	−0.0888959	−0.0444479	+	0.999012i	$0.514153\pi$
$402$	−2.06903e6	−0.638559
$403$	−790393.	−0.242427
$404$	1.68883e6	0.514793
$405$	0	0
$406$	−7.29756e6	−2.19716
$407$	−2.40136e6	−0.718572
$408$	577152.	0.171648
$409$	4.12069e6	1.21804	0.609019	−	0.793155i	$-0.291563\pi$
$409$	4.12069e6	1.21804	0.609019	+	0.793155i	$0.291563\pi$
$410$	0	0
$411$	−2.05810e6	−0.600983
$412$	2.83898e6	0.823984
$413$	−8.36703e6	−2.41377
$414$	−503496.	−0.144376
$415$	0	0
$416$	−828416.	−0.234701
$417$	3.05766e6	0.861091
$418$	3.39636e6	0.950765
$419$	−2.37948e6	−0.662136	−0.331068	−	0.943607i	$-0.607409\pi$
$419$	−2.37948e6	−0.662136	−0.331068	+	0.943607i	$0.607409\pi$
$420$	0	0
$421$	−741298.	−0.203839	−0.101920	−	0.994793i	$-0.532498\pi$
$421$	−741298.	−0.203839	−0.101920	+	0.994793i	$0.532498\pi$
$422$	−1.85069e6	−0.505886
$423$	−681858.	−0.185286
$424$	−1.13050e6	−0.305390
$425$	0	0
$426$	−271728.	−0.0725455
$427$	−821791.	−0.218118
$428$	2.94067e6	0.775956
$429$	3.62594e6	0.951212
$430$	0	0
$431$	−187398.	−0.0485928	−0.0242964	−	0.999705i	$-0.507735\pi$
$431$	−187398.	−0.0485928	−0.0242964	+	0.999705i	$0.507735\pi$
$432$	186624.	0.0481125
$433$	−6.55110e6	−1.67917	−0.839585	−	0.543229i	$-0.817202\pi$
$433$	−6.55110e6	−1.67917	−0.839585	+	0.543229i	$0.817202\pi$
$434$	−910564.	−0.232052
$435$	0	0
$436$	2.71496e6	0.683986
$437$	2.64957e6	0.663700
$438$	23256.0	0.00579228
$439$	270065.	0.0668817	0.0334408	−	0.999441i	$-0.489353\pi$
$439$	270065.	0.0668817	0.0334408	+	0.999441i	$0.489353\pi$
$440$	0	0
$441$	3.03604e6	0.743381
$442$	−3.24247e6	−0.789443
$443$	−2.77934e6	−0.672873	−0.336436	−	0.941706i	$-0.609222\pi$
$443$	−2.77934e6	−0.672873	−0.336436	+	0.941706i	$0.609222\pi$
$444$	694368.	0.167160
$445$	0	0
$446$	2.94108e6	0.700116
$447$	2.75805e6	0.652880
$448$	−954368.	−0.224657
$449$	7.86630e6	1.84143	0.920714	−	0.390238i	$-0.127607\pi$
$449$	7.86630e6	1.84143	0.920714	+	0.390238i	$0.127607\pi$
$450$	0	0
$451$	4.05770e6	0.939375
$452$	−4.21574e6	−0.970573
$453$	1.78944e6	0.409706
$454$	−3.86875e6	−0.880909
$455$	0	0
$456$	−982080.	−0.221174
$457$	−6.23356e6	−1.39619	−0.698097	−	0.716004i	$-0.745969\pi$
$457$	−6.23356e6	−1.39619	−0.698097	+	0.716004i	$0.745969\pi$
$458$	−334780.	−0.0745754
$459$	730458.	0.161832
$460$	0	0
$461$	−4.68305e6	−1.02630	−0.513152	−	0.858298i	$-0.671522\pi$
$461$	−4.68305e6	−1.02630	−0.513152	+	0.858298i	$0.671522\pi$
$462$	4.17722e6	0.910506
$463$	382816.	0.0829923	0.0414961	−	0.999139i	$-0.486788\pi$
$463$	382816.	0.0829923	0.0414961	+	0.999139i	$0.486788\pi$
$464$	2.00448e6	0.432222
$465$	0	0
$466$	3.49550e6	0.745667
$467$	1.93540e6	0.410657	0.205328	−	0.978693i	$-0.434174\pi$
$467$	1.93540e6	0.410657	0.205328	+	0.978693i	$0.434174\pi$
$468$	−1.04846e6	−0.221278
$469$	1.33912e7	2.81117
$470$	0	0
$471$	−3.25155e6	−0.675364
$472$	2.29824e6	0.474832
$473$	9.69556e6	1.99260
$474$	−817920.	−0.167211
$475$	0	0
$476$	−3.73546e6	−0.755660
$477$	−1.43078e6	−0.287924
$478$	−4.24224e6	−0.849230
$479$	−5.56917e6	−1.10905	−0.554526	−	0.832167i	$-0.687100\pi$
$479$	−5.56917e6	−1.10905	−0.554526	+	0.832167i	$0.687100\pi$
$480$	0	0
$481$	−3.90100e6	−0.768799
$482$	−3.02729e6	−0.593522
$483$	3.25874e6	0.635597
$484$	1.39125e6	0.269955
$485$	0	0
$486$	236196.	0.0453609
$487$	−4.22450e6	−0.807148	−0.403574	−	0.914947i	$-0.632232\pi$
$487$	−4.22450e6	−0.807148	−0.403574	+	0.914947i	$0.632232\pi$
$488$	225728.	0.0429078
$489$	−3.04343e6	−0.575561
$490$	0	0
$491$	6.96295e6	1.30344	0.651718	−	0.758461i	$-0.274049\pi$
$491$	6.96295e6	1.30344	0.651718	+	0.758461i	$0.274049\pi$
$492$	−1.17331e6	−0.218525
$493$	7.84566e6	1.45383
$494$	5.51738e6	1.01722
$495$	0	0
$496$	250112.	0.0456489
$497$	1.75868e6	0.319372
$498$	−416664.	−0.0752857
$499$	−9.29582e6	−1.67123	−0.835616	−	0.549315i	$-0.814889\pi$
$499$	−9.29582e6	−1.67123	−0.835616	+	0.549315i	$0.814889\pi$
$500$	0	0
$501$	−3.87223e6	−0.689235
$502$	−2.54059e6	−0.449962
$503$	−6.34136e6	−1.11754	−0.558770	−	0.829323i	$-0.688726\pi$
$503$	−6.34136e6	−1.11754	−0.558770	+	0.829323i	$0.688726\pi$
$504$	−1.20787e6	−0.211809
$505$	0	0
$506$	3.09557e6	0.537482
$507$	2.54869e6	0.440349
$508$	4.11059e6	0.706716
$509$	−7.38309e6	−1.26312	−0.631559	−	0.775328i	$-0.717585\pi$
$509$	−7.38309e6	−1.26312	−0.631559	+	0.775328i	$0.717585\pi$
$510$	0	0
$511$	−150518.	−0.0254998
$512$	262144.	0.0441942
$513$	−1.24294e6	−0.208525
$514$	−122832.	−0.0205071
$515$	0	0
$516$	−2.80354e6	−0.463534
$517$	4.19216e6	0.689782
$518$	−4.49410e6	−0.735900
$519$	−5.43019e6	−0.884904
$520$	0	0
$521$	4.19620e6	0.677270	0.338635	−	0.940918i	$-0.390035\pi$
$521$	4.19620e6	0.677270	0.338635	+	0.940918i	$0.390035\pi$
$522$	2.53692e6	0.407503
$523$	3.57942e6	0.572214	0.286107	−	0.958198i	$-0.407639\pi$
$523$	3.57942e6	0.572214	0.286107	+	0.958198i	$0.407639\pi$
$524$	−3.50477e6	−0.557611
$525$	0	0
$526$	758064.	0.119465
$527$	978954.	0.153545
$528$	−1.14739e6	−0.179113
$529$	−4.02143e6	−0.624800
$530$	0	0
$531$	2.90871e6	0.447676
$532$	6.35624e6	0.973691
$533$	6.59173e6	1.00504
$534$	−2.84256e6	−0.431377
$535$	0	0
$536$	−3.67827e6	−0.553009
$537$	−3.36933e6	−0.504206
$538$	4.43988e6	0.661326
$539$	−1.86660e7	−2.76745
$540$	0	0
$541$	4.95548e6	0.727934	0.363967	−	0.931412i	$-0.381422\pi$
$541$	4.95548e6	0.727934	0.363967	+	0.931412i	$0.381422\pi$
$542$	847808.	0.123965
$543$	−2.09181e6	−0.304454
$544$	1.02605e6	0.148652
$545$	0	0
$546$	6.78589e6	0.974149
$547$	5.31803e6	0.759946	0.379973	−	0.924998i	$-0.375933\pi$
$547$	5.31803e6	0.759946	0.379973	+	0.924998i	$0.375933\pi$
$548$	−3.65885e6	−0.520467
$549$	285687.	0.0404538
$550$	0	0
$551$	−1.33502e7	−1.87330
$552$	−895104.	−0.125033
$553$	5.29376e6	0.736125
$554$	−4.18965e6	−0.579967
$555$	0	0
$556$	5.43584e6	0.745727
$557$	4.25794e6	0.581516	0.290758	−	0.956797i	$-0.406093\pi$
$557$	4.25794e6	0.581516	0.290758	+	0.956797i	$0.406093\pi$
$558$	316548.	0.0430382
$559$	1.57504e7	2.13188
$560$	0	0
$561$	−4.49096e6	−0.602466
$562$	136008.	0.0181645
$563$	4.37617e6	0.581866	0.290933	−	0.956743i	$-0.406034\pi$
$563$	4.37617e6	0.581866	0.290933	+	0.956743i	$0.406034\pi$
$564$	−1.21219e6	−0.160463
$565$	0	0
$566$	1.12440e6	0.147530
$567$	−1.52871e6	−0.199696
$568$	−483072.	−0.0628262
$569$	3.57717e6	0.463190	0.231595	−	0.972812i	$-0.425606\pi$
$569$	3.57717e6	0.463190	0.231595	+	0.972812i	$0.425606\pi$
$570$	0	0
$571$	1.94693e6	0.249896	0.124948	−	0.992163i	$-0.460124\pi$
$571$	1.94693e6	0.249896	0.124948	+	0.992163i	$0.460124\pi$
$572$	6.44611e6	0.823773
$573$	−7.61578e6	−0.969009
$574$	7.59394e6	0.962027
$575$	0	0
$576$	331776.	0.0416667
$577$	−6.95576e6	−0.869772	−0.434886	−	0.900486i	$-0.643211\pi$
$577$	−6.95576e6	−0.869772	−0.434886	+	0.900486i	$0.643211\pi$
$578$	−1.66341e6	−0.207100
$579$	1.40023e6	0.173581
$580$	0	0
$581$	2.69674e6	0.331436
$582$	−1.96535e6	−0.240509
$583$	8.79667e6	1.07188
$584$	41344.0	0.00501626
$585$	0	0
$586$	6.23402e6	0.749936
$587$	−2.41853e6	−0.289705	−0.144852	−	0.989453i	$-0.546271\pi$
$587$	−2.41853e6	−0.289705	−0.144852	+	0.989453i	$0.546271\pi$
$588$	5.39741e6	0.643787
$589$	−1.66578e6	−0.197848
$590$	0	0
$591$	931338.	0.109683
$592$	1.23443e6	0.144765
$593$	9.58396e6	1.11920	0.559600	−	0.828763i	$-0.310955\pi$
$593$	9.58396e6	1.11920	0.559600	+	0.828763i	$0.310955\pi$
$594$	−1.45217e6	−0.168869
$595$	0	0
$596$	4.90320e6	0.565411
$597$	−1.26382e6	−0.145128
$598$	5.02874e6	0.575051
$599$	−1.52070e6	−0.173172	−0.0865858	−	0.996244i	$-0.527596\pi$
$599$	−1.52070e6	−0.173172	−0.0865858	+	0.996244i	$0.527596\pi$
$600$	0	0
$601$	3.88283e6	0.438492	0.219246	−	0.975670i	$-0.429640\pi$
$601$	3.88283e6	0.438492	0.219246	+	0.975670i	$0.429640\pi$
$602$	1.81451e7	2.04065
$603$	−4.65531e6	−0.521381
$604$	3.18123e6	0.354816
$605$	0	0
$606$	3.79987e6	0.420327
$607$	107992.	0.0118965	0.00594826	−	0.999982i	$-0.498107\pi$
$607$	107992.	0.0118965	0.00594826	+	0.999982i	$0.498107\pi$
$608$	−1.74592e6	−0.191543
$609$	−1.64195e7	−1.79398
$610$	0	0
$611$	6.81016e6	0.737997
$612$	1.29859e6	0.140150
$613$	−7.49923e6	−0.806057	−0.403028	−	0.915187i	$-0.632042\pi$
$613$	−7.49923e6	−0.806057	−0.403028	+	0.915187i	$0.632042\pi$
$614$	3.35967e6	0.359646
$615$	0	0
$616$	7.42618e6	0.788521
$617$	1.22695e6	0.129752	0.0648761	−	0.997893i	$-0.479335\pi$
$617$	1.22695e6	0.129752	0.0648761	+	0.997893i	$0.479335\pi$
$618$	6.38770e6	0.672780
$619$	9.51340e6	0.997950	0.498975	−	0.866616i	$-0.333710\pi$
$619$	9.51340e6	0.997950	0.498975	+	0.866616i	$0.333710\pi$
$620$	0	0
$621$	−1.13287e6	−0.117883
$622$	−1.17079e6	−0.121340
$623$	1.83977e7	1.89908
$624$	−1.86394e6	−0.191633
$625$	0	0
$626$	−511436.	−0.0521621
$627$	7.64181e6	0.776296
$628$	−5.78053e6	−0.584882
$629$	4.83164e6	0.486932
$630$	0	0
$631$	−1.56449e7	−1.56423	−0.782114	−	0.623135i	$-0.785859\pi$
$631$	−1.56449e7	−1.56423	−0.782114	+	0.623135i	$0.785859\pi$
$632$	−1.45408e6	−0.144809
$633$	−4.16406e6	−0.413055
$634$	−2.59219e6	−0.256120
$635$	0	0
$636$	−2.54362e6	−0.249350
$637$	−3.03229e7	−2.96089
$638$	−1.55974e7	−1.51705
$639$	−611388.	−0.0592331
$640$	0	0
$641$	−9.60395e6	−0.923219	−0.461609	−	0.887083i	$-0.652728\pi$
$641$	−9.60395e6	−0.923219	−0.461609	+	0.887083i	$0.652728\pi$
$642$	6.61651e6	0.633566
$643$	8.24396e6	0.786336	0.393168	−	0.919467i	$-0.371379\pi$
$643$	8.24396e6	0.786336	0.393168	+	0.919467i	$0.371379\pi$
$644$	5.79331e6	0.550443
$645$	0	0
$646$	−6.83364e6	−0.644274
$647$	4.07353e6	0.382570	0.191285	−	0.981535i	$-0.438735\pi$
$647$	4.07353e6	0.382570	0.191285	+	0.981535i	$0.438735\pi$
$648$	419904.	0.0392837
$649$	−1.78832e7	−1.66661
$650$	0	0
$651$	−2.04877e6	−0.189470
$652$	−5.41054e6	−0.498450
$653$	−1.68193e7	−1.54357	−0.771783	−	0.635886i	$-0.780635\pi$
$653$	−1.68193e7	−1.54357	−0.771783	+	0.635886i	$0.780635\pi$
$654$	6.10866e6	0.558472
$655$	0	0
$656$	−2.08589e6	−0.189248
$657$	52326.0	0.00472938
$658$	7.84558e6	0.706415
$659$	2.87826e6	0.258176	0.129088	−	0.991633i	$-0.458795\pi$
$659$	2.87826e6	0.258176	0.129088	+	0.991633i	$0.458795\pi$
$660$	0	0
$661$	1.33386e7	1.18743	0.593713	−	0.804677i	$-0.297661\pi$
$661$	1.33386e7	1.18743	0.593713	+	0.804677i	$0.297661\pi$
$662$	−1.35726e7	−1.20370
$663$	−7.29556e6	−0.644577
$664$	−740736.	−0.0651993
$665$	0	0
$666$	1.56233e6	0.136486
$667$	−1.21678e7	−1.05901
$668$	−6.88397e6	−0.596895
$669$	6.61744e6	0.571643
$670$	0	0
$671$	−1.75645e6	−0.150601
$672$	−2.14733e6	−0.183432
$673$	−6.37345e6	−0.542422	−0.271211	−	0.962520i	$-0.587424\pi$
$673$	−6.37345e6	−0.542422	−0.271211	+	0.962520i	$0.587424\pi$
$674$	6.48339e6	0.549734
$675$	0	0
$676$	4.53101e6	0.381354
$677$	2.27210e7	1.90526	0.952632	−	0.304126i	$-0.0983645\pi$
$677$	2.27210e7	1.90526	0.952632	+	0.304126i	$0.0983645\pi$
$678$	−9.48542e6	−0.792469
$679$	1.27202e7	1.05881
$680$	0	0
$681$	−8.70469e6	−0.719260
$682$	−1.94618e6	−0.160222
$683$	1.53612e7	1.26001	0.630003	−	0.776593i	$-0.283054\pi$
$683$	1.53612e7	1.26001	0.630003	+	0.776593i	$0.283054\pi$
$684$	−2.20968e6	−0.180588
$685$	0	0
$686$	−1.92691e7	−1.56333
$687$	−753255.	−0.0608906
$688$	−4.98406e6	−0.401432
$689$	1.42902e7	1.14680
$690$	0	0
$691$	8.05035e6	0.641386	0.320693	−	0.947183i	$-0.396084\pi$
$691$	8.05035e6	0.641386	0.320693	+	0.947183i	$0.396084\pi$
$692$	−9.65366e6	−0.766350
$693$	9.39875e6	0.743425
$694$	5.14553e6	0.405538
$695$	0	0
$696$	4.51008e6	0.352908
$697$	−8.16430e6	−0.636556
$698$	−1.09222e7	−0.848539
$699$	7.86488e6	0.608835
$700$	0	0
$701$	7.27840e6	0.559424	0.279712	−	0.960084i	$-0.409761\pi$
$701$	7.27840e6	0.559424	0.279712	+	0.960084i	$0.409761\pi$
$702$	−2.35904e6	−0.180673
$703$	−8.22151e6	−0.627427
$704$	−2.03981e6	−0.155116
$705$	0	0
$706$	8.54594e6	0.645280
$707$	−2.45936e7	−1.85044
$708$	5.17104e6	0.387699
$709$	1.37498e7	1.02726	0.513630	−	0.858012i	$-0.328300\pi$
$709$	1.37498e7	1.02726	0.513630	+	0.858012i	$0.328300\pi$
$710$	0	0
$711$	−1.84032e6	−0.136527
$712$	−5.05344e6	−0.373583
$713$	−1.51826e6	−0.111846
$714$	−8.40478e6	−0.616994
$715$	0	0
$716$	−5.98992e6	−0.436655
$717$	−9.54504e6	−0.693394
$718$	1.30562e7	0.945164
$719$	−9.05583e6	−0.653290	−0.326645	−	0.945147i	$-0.605918\pi$
$719$	−9.05583e6	−0.653290	−0.326645	+	0.945147i	$0.605918\pi$
$720$	0	0
$721$	−4.13426e7	−2.96183
$722$	1.72370e6	0.123061
$723$	−6.81141e6	−0.484609
$724$	−3.71877e6	−0.263665
$725$	0	0
$726$	3.13031e6	0.220417
$727$	2.20979e7	1.55065	0.775327	−	0.631560i	$-0.217585\pi$
$727$	2.20979e7	1.55065	0.775327	+	0.631560i	$0.217585\pi$
$728$	1.20638e7	0.843638
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.95079e7	−1.35026
$732$	507888.	0.0350340
$733$	−2.07337e6	−0.142534	−0.0712669	−	0.997457i	$-0.522704\pi$
$733$	−2.07337e6	−0.142534	−0.0712669	+	0.997457i	$0.522704\pi$
$734$	1.66031e7	1.13749
$735$	0	0
$736$	−1.59130e6	−0.108282
$737$	2.86216e7	1.94100
$738$	−2.63995e6	−0.178425
$739$	1.48849e7	1.00262	0.501310	−	0.865268i	$-0.332852\pi$
$739$	1.48849e7	1.00262	0.501310	+	0.865268i	$0.332852\pi$
$740$	0	0
$741$	1.24141e7	0.830558
$742$	1.64628e7	1.09773
$743$	−2.58856e7	−1.72023	−0.860116	−	0.510099i	$-0.829609\pi$
$743$	−2.58856e7	−1.72023	−0.860116	+	0.510099i	$0.829609\pi$
$744$	562752.	0.0372722
$745$	0	0
$746$	8.56968e6	0.563790
$747$	−937494.	−0.0614705
$748$	−7.98394e6	−0.521751
$749$	−4.28235e7	−2.78919
$750$	0	0
$751$	−7.98645e6	−0.516718	−0.258359	−	0.966049i	$-0.583182\pi$
$751$	−7.98645e6	−0.516718	−0.258359	+	0.966049i	$0.583182\pi$
$752$	−2.15501e6	−0.138965
$753$	−5.71633e6	−0.367392
$754$	−2.53379e7	−1.62309
$755$	0	0
$756$	−2.71771e6	−0.172941
$757$	1.15570e7	0.733000	0.366500	−	0.930418i	$-0.380556\pi$
$757$	1.15570e7	0.733000	0.366500	+	0.930418i	$0.380556\pi$
$758$	−7.78798e6	−0.492325
$759$	6.96503e6	0.438852
$760$	0	0
$761$	3.16705e6	0.198241	0.0991205	−	0.995075i	$-0.468397\pi$
$761$	3.16705e6	0.198241	0.0991205	+	0.995075i	$0.468397\pi$
$762$	9.24883e6	0.577031
$763$	−3.95366e7	−2.45860
$764$	−1.35392e7	−0.839187
$765$	0	0
$766$	−1.06021e7	−0.652860
$767$	−2.90512e7	−1.78310
$768$	589824.	0.0360844
$769$	8.21560e6	0.500983	0.250492	−	0.968119i	$-0.419408\pi$
$769$	8.21560e6	0.500983	0.250492	+	0.968119i	$0.419408\pi$
$770$	0	0
$771$	−276372.	−0.0167440
$772$	2.48930e6	0.150326
$773$	−1.19708e7	−0.720567	−0.360284	−	0.932843i	$-0.617320\pi$
$773$	−1.19708e7	−0.720567	−0.360284	+	0.932843i	$0.617320\pi$
$774$	−6.30796e6	−0.378474
$775$	0	0
$776$	−3.49395e6	−0.208287
$777$	−1.01117e7	−0.600860
$778$	1.13016e6	0.0669408
$779$	1.38923e7	0.820223
$780$	0	0
$781$	3.75890e6	0.220513
$782$	−6.22843e6	−0.364218
$783$	5.70807e6	0.332725
$784$	9.59539e6	0.557536
$785$	0	0
$786$	−7.88573e6	−0.455287
$787$	1.71154e7	0.985032	0.492516	−	0.870304i	$-0.336077\pi$
$787$	1.71154e7	0.985032	0.492516	+	0.870304i	$0.336077\pi$
$788$	1.65571e6	0.0949881
$789$	1.70564e6	0.0975429
$790$	0	0
$791$	6.13918e7	3.48874
$792$	−2.58163e6	−0.146245
$793$	−2.85334e6	−0.161128
$794$	1.04826e7	0.590091
$795$	0	0
$796$	−2.24680e6	−0.125685
$797$	2.80753e7	1.56559	0.782797	−	0.622277i	$-0.213792\pi$
$797$	2.80753e7	1.56559	0.782797	+	0.622277i	$0.213792\pi$
$798$	1.43015e7	0.795015
$799$	−8.43484e6	−0.467423
$800$	0	0
$801$	−6.39576e6	−0.352217
$802$	−1.14499e6	−0.0628589
$803$	−321708.	−0.0176065
$804$	−8.27611e6	−0.451530
$805$	0	0
$806$	−3.16157e6	−0.171422
$807$	9.98973e6	0.539970
$808$	6.75533e6	0.364014
$809$	9.90816e6	0.532257	0.266129	−	0.963938i	$-0.414255\pi$
$809$	9.90816e6	0.532257	0.266129	+	0.963938i	$0.414255\pi$
$810$	0	0
$811$	5.72573e6	0.305688	0.152844	−	0.988250i	$-0.451157\pi$
$811$	5.72573e6	0.305688	0.152844	+	0.988250i	$0.451157\pi$
$812$	−2.91902e7	−1.55363
$813$	1.90757e6	0.101217
$814$	−9.60542e6	−0.508107
$815$	0	0
$816$	2.30861e6	0.121374
$817$	3.31946e7	1.73985
$818$	1.64827e7	0.861284
$819$	1.52683e7	0.795389
$820$	0	0
$821$	−5.96570e6	−0.308890	−0.154445	−	0.988001i	$-0.549359\pi$
$821$	−5.96570e6	−0.308890	−0.154445	+	0.988001i	$0.549359\pi$
$822$	−8.23241e6	−0.424959
$823$	1.97778e7	1.01784	0.508918	−	0.860815i	$-0.330046\pi$
$823$	1.97778e7	1.01784	0.508918	+	0.860815i	$0.330046\pi$
$824$	1.13559e7	0.582645
$825$	0	0
$826$	−3.34681e7	−1.70679
$827$	−2.04045e7	−1.03744	−0.518719	−	0.854945i	$-0.673591\pi$
$827$	−2.04045e7	−1.03744	−0.518719	+	0.854945i	$0.673591\pi$
$828$	−2.01398e6	−0.102089
$829$	−2.77183e7	−1.40081	−0.700406	−	0.713745i	$-0.746998\pi$
$829$	−2.77183e7	−1.40081	−0.700406	+	0.713745i	$0.746998\pi$
$830$	0	0
$831$	−9.42672e6	−0.473541
$832$	−3.31366e6	−0.165959
$833$	3.75570e7	1.87533
$834$	1.22306e7	0.608883
$835$	0	0
$836$	1.35854e7	0.672292
$837$	712233.	0.0351405
$838$	−9.51792e6	−0.468201
$839$	−7.79841e6	−0.382473	−0.191237	−	0.981544i	$-0.561250\pi$
$839$	−7.79841e6	−0.382473	−0.191237	+	0.981544i	$0.561250\pi$
$840$	0	0
$841$	4.07978e7	1.98905
$842$	−2.96519e6	−0.144136
$843$	306018.	0.0148313
$844$	−7.40277e6	−0.357716
$845$	0	0
$846$	−2.72743e6	−0.131017
$847$	−2.02600e7	−0.970358
$848$	−4.52198e6	−0.215943
$849$	2.52991e6	0.120458
$850$	0	0
$851$	−7.49339e6	−0.354694
$852$	−1.08691e6	−0.0512974
$853$	−6.22554e6	−0.292957	−0.146479	−	0.989214i	$-0.546794\pi$
$853$	−6.22554e6	−0.292957	−0.146479	+	0.989214i	$0.546794\pi$
$854$	−3.28716e6	−0.154233
$855$	0	0
$856$	1.17627e7	0.548684
$857$	−3.39757e7	−1.58022	−0.790108	−	0.612968i	$-0.789976\pi$
$857$	−3.39757e7	−1.58022	−0.790108	+	0.612968i	$0.789976\pi$
$858$	1.45038e7	0.672608
$859$	−1.47282e7	−0.681033	−0.340516	−	0.940239i	$-0.610602\pi$
$859$	−1.47282e7	−0.681033	−0.340516	+	0.940239i	$0.610602\pi$
$860$	0	0
$861$	1.70864e7	0.785492
$862$	−749592.	−0.0343603
$863$	1.88594e7	0.861988	0.430994	−	0.902355i	$-0.358163\pi$
$863$	1.88594e7	0.861988	0.430994	+	0.902355i	$0.358163\pi$
$864$	746496.	0.0340207
$865$	0	0
$866$	−2.62044e7	−1.18735
$867$	−3.74268e6	−0.169096
$868$	−3.64226e6	−0.164086
$869$	1.13146e7	0.508263
$870$	0	0
$871$	4.64957e7	2.07667
$872$	1.08598e7	0.483651
$873$	−4.42203e6	−0.196375
$874$	1.05983e7	0.469307
$875$	0	0
$876$	93024.0	0.00409576
$877$	1.82112e7	0.799540	0.399770	−	0.916615i	$-0.369090\pi$
$877$	1.82112e7	0.799540	0.399770	+	0.916615i	$0.369090\pi$
$878$	1.08026e6	0.0472925
$879$	1.40266e7	0.612321
$880$	0	0
$881$	−7.65425e6	−0.332248	−0.166124	−	0.986105i	$-0.553125\pi$
$881$	−7.65425e6	−0.332248	−0.166124	+	0.986105i	$0.553125\pi$
$882$	1.21442e7	0.525650
$883$	597451.	0.0257870	0.0128935	−	0.999917i	$-0.495896\pi$
$883$	597451.	0.0257870	0.0128935	+	0.999917i	$0.495896\pi$
$884$	−1.29699e7	−0.558220
$885$	0	0
$886$	−1.11174e7	−0.475793
$887$	−2.06888e6	−0.0882929	−0.0441465	−	0.999025i	$-0.514057\pi$
$887$	−2.06888e6	−0.0882929	−0.0441465	+	0.999025i	$0.514057\pi$
$888$	2.77747e6	0.118200
$889$	−5.98605e7	−2.54031
$890$	0	0
$891$	−3.26738e6	−0.137881
$892$	1.17643e7	0.495057
$893$	1.43527e7	0.602289
$894$	1.10322e7	0.461656
$895$	0	0
$896$	−3.81747e6	−0.158857
$897$	1.13147e7	0.469527
$898$	3.14652e7	1.30209
$899$	7.64991e6	0.315687
$900$	0	0
$901$	−1.76993e7	−0.726348
$902$	1.62308e7	0.664238
$903$	4.08265e7	1.66618
$904$	−1.68630e7	−0.686299
$905$	0	0
$906$	7.15777e6	0.289706
$907$	−7.83331e6	−0.316175	−0.158087	−	0.987425i	$-0.550533\pi$
$907$	−7.83331e6	−0.316175	−0.158087	+	0.987425i	$0.550533\pi$
$908$	−1.54750e7	−0.622897
$909$	8.54971e6	0.343196
$910$	0	0
$911$	4.08133e7	1.62932	0.814659	−	0.579940i	$-0.196924\pi$
$911$	4.08133e7	1.62932	0.814659	+	0.579940i	$0.196924\pi$
$912$	−3.92832e6	−0.156394
$913$	5.76385e6	0.228842
$914$	−2.49342e7	−0.987258
$915$	0	0
$916$	−1.33912e6	−0.0527328
$917$	5.10382e7	2.00434
$918$	2.92183e6	0.114432
$919$	6.21579e6	0.242777	0.121389	−	0.992605i	$-0.461265\pi$
$919$	6.21579e6	0.242777	0.121389	+	0.992605i	$0.461265\pi$
$920$	0	0
$921$	7.55925e6	0.293650
$922$	−1.87322e7	−0.725707
$923$	6.10633e6	0.235926
$924$	1.67089e7	0.643825
$925$	0	0
$926$	1.53126e6	0.0586844
$927$	1.43723e7	0.549323
$928$	8.01792e6	0.305627
$929$	−4.64847e6	−0.176714	−0.0883570	−	0.996089i	$-0.528162\pi$
$929$	−4.64847e6	−0.176714	−0.0883570	+	0.996089i	$0.528162\pi$
$930$	0	0
$931$	−6.39068e7	−2.41642
$932$	1.39820e7	0.527266
$933$	−2.63428e6	−0.0990736
$934$	7.74161e6	0.290378
$935$	0	0
$936$	−4.19386e6	−0.156467
$937$	1.33603e7	0.497127	0.248563	−	0.968616i	$-0.420042\pi$
$937$	1.33603e7	0.497127	0.248563	+	0.968616i	$0.420042\pi$
$938$	5.35648e7	1.98780
$939$	−1.15073e6	−0.0425902
$940$	0	0
$941$	3.32569e7	1.22436	0.612178	−	0.790720i	$-0.290294\pi$
$941$	3.32569e7	1.22436	0.612178	+	0.790720i	$0.290294\pi$
$942$	−1.30062e7	−0.477554
$943$	1.26620e7	0.463685
$944$	9.19296e6	0.335757
$945$	0	0
$946$	3.87822e7	1.40898
$947$	−4.21530e7	−1.52740	−0.763702	−	0.645569i	$-0.776620\pi$
$947$	−4.21530e7	−1.52740	−0.763702	+	0.645569i	$0.776620\pi$
$948$	−3.27168e6	−0.118236
$949$	−522614.	−0.0188372
$950$	0	0
$951$	−5.83243e6	−0.209121
$952$	−1.49418e7	−0.534332
$953$	−2.24691e7	−0.801406	−0.400703	−	0.916208i	$-0.631234\pi$
$953$	−2.24691e7	−0.801406	−0.400703	+	0.916208i	$0.631234\pi$
$954$	−5.72314e6	−0.203593
$955$	0	0
$956$	−1.69690e7	−0.600497
$957$	−3.50941e7	−1.23866
$958$	−2.22767e7	−0.784218
$959$	5.32820e7	1.87083
$960$	0	0
$961$	−2.76746e7	−0.966659
$962$	−1.56040e7	−0.543623
$963$	1.48872e7	0.517304
$964$	−1.21092e7	−0.419683
$965$	0	0
$966$	1.30350e7	0.449435
$967$	1.75000e7	0.601826	0.300913	−	0.953652i	$-0.402709\pi$
$967$	1.75000e7	0.601826	0.300913	+	0.953652i	$0.402709\pi$
$968$	5.56499e6	0.190887
$969$	−1.53757e7	−0.526048
$970$	0	0
$971$	5.42920e7	1.84794	0.923970	−	0.382465i	$-0.124925\pi$
$971$	5.42920e7	1.84794	0.923970	+	0.382465i	$0.124925\pi$
$972$	944784.	0.0320750
$973$	−7.91594e7	−2.68053
$974$	−1.68980e7	−0.570740
$975$	0	0
$976$	902912.	0.0303404
$977$	−2.55925e7	−0.857782	−0.428891	−	0.903356i	$-0.641096\pi$
$977$	−2.55925e7	−0.857782	−0.428891	+	0.903356i	$0.641096\pi$
$978$	−1.21737e7	−0.406983
$979$	3.93221e7	1.31123
$980$	0	0
$981$	1.37445e7	0.455991
$982$	2.78518e7	0.921668
$983$	4.82488e6	0.159258	0.0796292	−	0.996825i	$-0.474626\pi$
$983$	4.82488e6	0.159258	0.0796292	+	0.996825i	$0.474626\pi$
$984$	−4.69325e6	−0.154520
$985$	0	0
$986$	3.13826e7	1.02801
$987$	1.76525e7	0.576786
$988$	2.20695e7	0.719284
$989$	3.02548e7	0.983567
$990$	0	0
$991$	1.18448e6	0.0383127	0.0191563	−	0.999817i	$-0.493902\pi$
$991$	1.18448e6	0.0383127	0.0191563	+	0.999817i	$0.493902\pi$
$992$	1.00045e6	0.0322786
$993$	−3.05383e7	−0.982816
$994$	7.03474e6	0.225830
$995$	0	0
$996$	−1.66666e6	−0.0532350
$997$	2.58178e7	0.822585	0.411293	−	0.911503i	$-0.365077\pi$
$997$	2.58178e7	0.822585	0.411293	+	0.911503i	$0.365077\pi$
$998$	−3.71833e7	−1.18174
$999$	3.51524e6	0.111440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.a.l.1.1	yes	1
3.2	odd	2		450.6.a.a.1.1		1
5.2	odd	4		150.6.c.e.49.2		2
5.3	odd	4		150.6.c.e.49.1		2
5.4	even	2		150.6.a.c.1.1	✓	1
15.2	even	4		450.6.c.n.199.1		2
15.8	even	4		450.6.c.n.199.2		2
15.14	odd	2		450.6.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.a.c.1.1	✓	1	5.4	even	2
150.6.a.l.1.1	yes	1	1.1	even	1	trivial
150.6.c.e.49.1		2	5.3	odd	4
150.6.c.e.49.2		2	5.2	odd	4
450.6.a.a.1.1		1	3.2	odd	2
450.6.a.x.1.1		1	15.14	odd	2
450.6.c.n.199.1		2	15.2	even	4
450.6.c.n.199.2		2	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more